- Email: [email protected]
Simulation of sub-femtosecond response in laser-assisted field emission
ARTICLE IN PRESS
Ultramicroscopy 107 (2007) 849–853 www.elsevier.com/locate/ultramic
Simulation of sub-femtosecond response in laser-assisted field emission M.J. Hagmanna,, M.S. Mousab a
NewPath Research, P.O. Box 3863, Salt Lake City, UT 84110, USA Department of Physics, Mu’tah University, P.O. Box 7, Al-Karak, Jordan
b
Abstract Numerical methods are used to simulate the response of field emission to incident pulses of optical radiation, with the objective of determining the criteria to reduce the time for response of the emitted current to the radiation. The results of these simulations suggest that a sub-femtosecond response may be achieved by increasing the power flux density of the radiation, and decreasing the applied static field. An intrinsic delay in the response is shown to correspond to the semiclassical time for traversal of the barrier by quantum tunneling. r 2007 Elsevier B.V. All rights reserved. PACS: 79.70; 02.60; 78.47 Keywords: Field emission; Femtosecond techniques; Numerical simulation; Tunneling time
1. Introduction Femtosecond (fs) science holds promise for realizing the dream of achieving spatial and temporal resolution on the atomic scale. For example, it may be possible to observe the motion of electrons in the valence states of molecules, allowing electronic processes such as charge transfer in complex molecules to be time resolved [1]. The first method proposed to create sub-fs pulses was a synthesizer in which two phase-locked lasers would generate a comb of six phase coherent signals that would be superimposed to generate the train of pulses [2]. In 1993, Macklin et al. [3] excited a thin layer of neon gas with 125 fs, 1015 W/cm2 pulses from a Ti:sapphire laser (l ¼ 800 nm) to generate a comb of high-order harmonics up to order 109 (l ¼ 7.3 nm, E ¼ 169 eV). Corkum et al. [1] explained the mechanism by which such high-order harmonics are produced, and proposed how to combine these harmonics to obtain a train of pulses as short as 10 attoseconds (as). LopezMartens et al. [4] described the experimental generation, compression, and delivery on target of ultrashort (170 as) pulses, generated by combining high-order harmonics that Corresponding author. Tel.: +1 801 573 9853.
E-mail address: [email protected] (M.J. Hagmann). 0304-3991/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.02.018
were generated by exciting argon atoms with 40 fs, 1014 W/ cm2 pulses from a Ti:sapphire laser (l ¼ 800 nm). Laser-assisted (or photon-assisted) field emission has recently been applied as a new technique for producing microwave radiation, and also shows promise as a new type of source for terahertz radiation [5]. In field emission, a strong static field bends the potential at the surface of a nanoscale tip so that electrons are emitted by quantum tunneling. The apex of the tip is much smaller than the wavelength of the optical radiation, so the time-dependent electric field of the radiation is superimposed on the applied static field to modulate the current of the emitted electrons. The nonlinear current–voltage characteristics of field emission cause the emitted current to have components at frequencies that are harmonics and mixer terms relative to the incident optical radiation. The tip can be used as an antenna or transmission line to transmit electromagnetic radiation at one or more of the frequencies which are present in the current. Now, we describe simulations of laser-assisted field emission with pulsed optical radiation to determine the effects of the experimental parameters on the time for the emitted current to respond to the pulsed radiation. These simulations were made to determine the feasibility of obtaining a sub-femtosecond response—a speed three
ARTICLE IN PRESS 850
M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
orders of magnitude greater than what we have previously considered. 2. Numerical methods for simulations We have developed two different numerical methods for simulating laser-assisted field emission with pulsed optical radiation [6,7]. It is well known that simple attempts to extend the finite difference techniques, which are satisfactory for solving the time-independent Schro¨dinger equation, become numerically unstable when they are applied to solve the time-dependent Schro¨dinger equation [8]. The cause for this instability is that the numerical approximations of the operators are not unitary even though the exact operators are unitary. Split-operator formulations have been used to obtain numerical stability [9], but then the solution is only implicit so that it is necessary to form and invert a large matrix. A variety of other techniques have also been developed to achieve numerical stability [10,11] but they all have much lower computational efficiency than a direct explicit solution. Numerical implementations often require empirically adjusted ‘‘windowing’’ and ‘‘gobbling’’ to achieve numerical stability [12] and absorbing boundary conditions, in which complex potentials are added at the boundaries to limit unphysical reflected waves originating at the boundaries of the grid [13]. Both of our new procedures are numerically stable, and neither requires the use of split-operators, windowing, gobbling, or absorbing boundary conditions. In the first procedure [6], the solution for the wave function is written as the product of the solution of the time-independent Schro¨dinger equation with the pulsed radiation field deleted, and an unknown function of the spatial coordinates and time. This form for the solution is substituted into the time-dependent Schro¨dinger equation to obtain a simpler equation that is solved using finite differences to determine the unknown product function. In the second procedure [7], the solution for the wave function is written as the sum of three terms. Each of these terms contains a different unknown function of the spatial coordinates and time. The second and third terms also have products of ejot and ejot, respectively, where o is the fundamental frequency of the sinusoid that is pulsed. This solution is substituted into the time-dependent Schro¨dinger equation, which is then partitioned to give three simpler equations, which are solved using finite differences to determine the three unknown functions. The simulations in the following section of this paper were made by using the second numerical method [7] with the Fowler–Nordheim model [14] for the potential at the surface of tungsten metal. 3. Results of simulations Figs. 1(a)–(f) show the probability density for electrons near a half-space of tungsten as a function of the distance from the surface (0–3.5 nm) and the time
(0–10 fs) for laser-assisted field emission. In each of these figures ES represents the applied electrostatic field, l and P are the wavelength and power flux density of the optical radiation, and ED is the RMS value of the dynamic electric field of the radiation. While the trajectories of the electrons are not defined, one may consider the movement of the electrons by viewing the planes for consecutively larger values of time in each of these six figures. Values of the current, corresponding to the number of electrons passing a point in one second, may also be determined from these figures. Two different effects of the optical radiation may be seen in the figures. First, the dynamic electric field of the optical radiation is superimposed on the static potential barrier to cause an effective barrier height that varies to follow each cycle of the radiation. Secondly, quanta may be absorbed or emitted by the electrons to cause sudden changes in the electron energy. A sinusoidal optical pulse at a wavelength l ¼ 400 nm is turned on at the time t ¼ 0. The applied static field ES ¼ 5.5 V/nm, and the radiation field has a power flux density P ¼ 1012, 1013, and 1014 W/m2 in Figs. 1(a)–(c), respectively. These three figures show that as the power flux density is increased, there is a change in the character of the response by the electrons. For a power flux density of 1012 W/m2 or less, the magnitude of the current follows the instantaneous value of the radiation field. That is, during each cycle of the optical radiation, the current is increased when the effective height of the barrier is lowered by superimposing the electric field of the optical radiation, and decreased when the effective height of the barrier is raised. However, for a power flux density of 1014 W/m2 or greater, a massive burst of current is caused by the optical radiation, which we refer to as ‘‘catastrophic laser-assisted field emission’’. A transition between these two types of response occurs at a power flux density of 1013 W/m2, where both of these aspects are present. Figs. 1(a)–(c) also show that, even though the sine wave for the incident optical pulse starts at time t ¼ 0, there is a dead-time before this radiation causes the height of any part of the surface to be above the curve that represents the probability density for t ¼ 0. In Section 4 we show that this dead-time is associated with the duration of quantum tunneling. These three figures show that the dead-time is reduced and the rate of rise for the current is increased by increasing the power flux density. The threshold for laser ablation with thin tungsten films corresponds to a fluence of 400 J/m2 [15]. We have used a power flux density of 1016 W/m2 in the simulations for Figs. 1(d)–(f) because this is below the threshold for ablation with pulse durations of 40 fs or less. However, it is possible that the tip could still be destroyed by explosive field emission because of the additional effects of the applied static field and the resulting current. Figs. 1(d)–(f) show the probability density for electrons near a half-space of tungsten when the sinusoidal optical pulse has a power flux density P ¼ 1016 W/m2. The applied
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
851
Fig. 1. Probability density of electrons near a planar surface of tungsten (a) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1012 W/m2, ED ¼ 0.027 V/nm rms. (b) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1013 W/m2, ED ¼ 0.087 V/nm rms. (c) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1014 W/m2, ED ¼ 0.27 V/nm rms. (d) ES ¼ 5.5 V/ nm, l ¼ 400 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms. (e) ES ¼ 3.0 V/nm, l ¼ 400 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms. (f) ES ¼ 3.0 V/nm, l ¼ 1000 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms.
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
852
static field ES ¼ 5.5, 3.0, and 3.0 V/nm, and the wavelength of the optical radiation l ¼ 400, 400, and 1000 nm, respectively, in Figs. 1(d)–(f). 4. Discussion 4.1. Relevance of tunneling time Measurements of the tunnel conductance in semiconductor heterostructures [16] and experiments with Josephson junctions [17] suggest that a specific time is required for the process of quantum tunneling. The duration of quantum tunneling has considerable practical significance. For example, measurements of the tunnel conductance in heterostructures differs from theory by as much as two orders of magnitude unless the image corrections are adjusted to allow for the traversal time [18]. Many different theoretical procedures have been used to determine the time that is required to traverse a barrier by quantum tunneling [19], such as in field emission. While there is not yet a complete consensus, 12 of these methods of analysis give the same result, corresponding to the imaginary value for the time that would be calculated for this process using classical physics, which we refer to as the ‘‘semiclassical traversal time’’ [20]. Furthermore, this result is consistent with experimental measurements of the tunnel conductance in heterostructures [21]. The semiclassical traversal time is given by Z z2 dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , T SC ¼ (1) 2 z1 m ðV ðzÞ E Þ where the particle with mass m and energy E tunnels through a potential V(z) from z1 to z2 [20]. Using the Fowler–Nordheim approximation for the potential in field emission [14] in Eq. (1), it may be shown that 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2T 0 F 0 F B 2 1 F0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiA, T SC ¼ (2) 1 þ 1 E@ F0 p F 1þ 1 F F0
where T 0 ¼
pffiffiffiffi 2 me 80 f3=2
(3)
4p0 f2 . (4) e3 Here e is the charge of an electron, j is the work function of the metal, e0 is the permittivity of free-space, F is the applied static field, and E(arg) is the Elliptic Integral of the Second Kind [22]. For Foo F0, which is a good approximation in field emission, these equations may be simplified to give 2T 0 F 0 T SC . (5) p F and
F0 ¼
Eq. (2) shows that for field emission with tungsten, the semiclassical traversal time is 1.4 fs for an applied static
field of 5.5 V/nm. This is in good agreement with the initial dead-time of approximately 1.5 fs that is seen in Fig. 1(a). We have already noted that the current is alternately increased and decreased during each half-cycle as the effective height of the barrier is lowered and raised by the optical radiation. However, it may be seen in Fig. 1(a) that each of these responses is delayed by approximately 0.7 fs, which is one-half of the semiclassical traversal time. 4.2. Catastrophic laser-assisted field emission By examining the data in Figs. 1(a)–(f), it may be seen that the initial dead-time is 1.5 fs for 1012 W/m2, 1.2 fs for 1013 W/m2, 0.8 fs for 1014 W/m2, and 0.4 fs in the three figures with 1016 W/m2, which shows that the dead-time is reduced as the power flux density is increased. Furthermore, these figures show that the current is greatly increased when the power flux density exceeds 1013 W/m2, which we refer to as ‘‘catastrophic laser-assisted field emission’’. Figs. 1(e) and (f) show that the power flux density has a much greater affect than the wavelength when using short pulses, and Figs. 1(d) and (e) show that reducing the applied static field causes a much sharper increase in the current. These phenomena may be understood in terms of two types of interaction: (1) the superposition of the electric field of the optical radiation on the potential barrier, and (2) the absorption and emission of quanta by the electrons. 5. Conclusions The simulations show that the dead-time before the current responds to the incident radiation may be reduced to 0.4 fs by using a greater power flux density of 1016 W/m2, where the current is increased to the point that we call ‘‘catastrophic laser-assisted field emission’’. This is several orders of magnitude faster than the phenomena that we have considered before [5], and these results suggest that it may be possible to use this method for an ultrafast optoelectronic switch. We have also found that the dead-time, as well as other delays in the response to the radiation, may be explained in terms of the semiclassical traversal time for quantum tunneling within the potential barrier. Experiments that are pertinent to these simulations are now in progress. Others have generated electron pulses with durations of tens of fs by irradiating a metal field emission tip with a fs laser [23,24], and they have reported a transition from photofield emission to optical field emission that is similar to the one that we have shown in these simulations at a much shorter time scale [23]. Analyses of recent measurements in which sub-8 fs lasers irradiated tungsten field emission tips show that this technique has promise for producing a single electron pulse with a duration of less than 1 fs [25]. Our simulations have the delimitations of (1) simulating the response of a half-space of metal so that three-dimensional
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
effects are not included, and (2) simulating the incident radiation as a sinusoidal field that is switched on whereas others have used the model of a wave packet with a Gaussian envelope which more accurately models the propagation of a pulse.
[8] [9] [10] [11] [12]
Acknowledgment
[13] [14]
The authors are grateful to Dr. E. P. Sheshin and Dr. A. S. Baturin, both of the Moscow Institute of Physics and Technology, Moscow, Russia, and M. Brugat, of Image Instrumentation Inc., Cooper City, FL, USA, for their advice concerning the technology of field emission. References [1] [2] [3] [4] [5] [6] [7]
P.B. Corkum, et al., Opt. Lett. 19 (1994) 1870. T.W. Ha¨nsch, Opt. Commun. 80 (1990) 71. J.J. Macklin, et al., Phys. Rev. Lett. 70 (1993) 766. R. Lopez-Martens, et al., Phys. Rev. Lett. 94 (2005) 33001. M.J. Hagmann, IEEE Trans. Microw. Theory Tech. 52 (2004) 236. M.J. Hagmann, Int. J. Quant. Chem. 60 (1996) 1231. M.J. Hagmann, Int. J. Quant. Chem. 70 (1998) 703.
[15] [16] [17] [18] [19] [20] [21]
[22]
[23] [24] [25]
853
A. Goldberg, et al., Am. J. Phys. 35 (1967) 177. K.C. Kulander, Phys. Rev. A 35 (1987) 445. C.W. McCurdy, et al., Phys. Rev. A 43 (1991) 5980. M.R. Hermann, J.A. Fleck Jr., Phys. Rev. A 38 (1988) 6000. J.T. Muckerman, et al., in: C. Cerjan (Ed.), Numerical Grid Methods and their Applications to Schro¨dinger’s Equation, Academic, Dordrecht, 1992. A. Vibok, G.G. Balint-Kurti, J. Chem. Phys. 96 (1992) 7615. R. Gomer, Field Emission and Field Ionization, American Institute of Physics, New York, 1993, pp. 53–54. Z. Toth, et al., Appl. Phys. A: Mat. Sci. Process. 60 (1995) 431. P. Gueret, et al., Appl. Phys. Lett. 53 (1988) 1617. D. Esteve, et al., Phys. Scr. T 29 (1989) 121. P. Gueret, et al., Electron. Lett. 21 (1985) 344. R. Landauer, Th. Martin, Rev. Mod. Phys. 66 (1994) 217. M.J. Hagmann, Int. J. Quant. Chem. 44 (1992) 299. P. Gueret, in: J.M. Chamberlain, et al., (Eds.), Electronic Properties of Multilayers and Low-Dimensional Semiconductor Structures, Plenum, New York, 1990, pp. 317–329. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government Printing Office, Washington, 1972, pp. 589–592. P. Hommelhoff, et al., Phys. Rev. Lett. 96 (2006) 077401. B.N. Mironov, et al., JETP Lett. 83 (2006) 369. P. Hommelhoff, et al., arXiv:quant-ph/0607006 v1 2 Jul 2006.
Ultramicroscopy 107 (2007) 849–853 www.elsevier.com/locate/ultramic
Simulation of sub-femtosecond response in laser-assisted field emission M.J. Hagmanna,, M.S. Mousab a
NewPath Research, P.O. Box 3863, Salt Lake City, UT 84110, USA Department of Physics, Mu’tah University, P.O. Box 7, Al-Karak, Jordan
b
Abstract Numerical methods are used to simulate the response of field emission to incident pulses of optical radiation, with the objective of determining the criteria to reduce the time for response of the emitted current to the radiation. The results of these simulations suggest that a sub-femtosecond response may be achieved by increasing the power flux density of the radiation, and decreasing the applied static field. An intrinsic delay in the response is shown to correspond to the semiclassical time for traversal of the barrier by quantum tunneling. r 2007 Elsevier B.V. All rights reserved. PACS: 79.70; 02.60; 78.47 Keywords: Field emission; Femtosecond techniques; Numerical simulation; Tunneling time
1. Introduction Femtosecond (fs) science holds promise for realizing the dream of achieving spatial and temporal resolution on the atomic scale. For example, it may be possible to observe the motion of electrons in the valence states of molecules, allowing electronic processes such as charge transfer in complex molecules to be time resolved [1]. The first method proposed to create sub-fs pulses was a synthesizer in which two phase-locked lasers would generate a comb of six phase coherent signals that would be superimposed to generate the train of pulses [2]. In 1993, Macklin et al. [3] excited a thin layer of neon gas with 125 fs, 1015 W/cm2 pulses from a Ti:sapphire laser (l ¼ 800 nm) to generate a comb of high-order harmonics up to order 109 (l ¼ 7.3 nm, E ¼ 169 eV). Corkum et al. [1] explained the mechanism by which such high-order harmonics are produced, and proposed how to combine these harmonics to obtain a train of pulses as short as 10 attoseconds (as). LopezMartens et al. [4] described the experimental generation, compression, and delivery on target of ultrashort (170 as) pulses, generated by combining high-order harmonics that Corresponding author. Tel.: +1 801 573 9853.
E-mail address: [email protected] (M.J. Hagmann). 0304-3991/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2007.02.018
were generated by exciting argon atoms with 40 fs, 1014 W/ cm2 pulses from a Ti:sapphire laser (l ¼ 800 nm). Laser-assisted (or photon-assisted) field emission has recently been applied as a new technique for producing microwave radiation, and also shows promise as a new type of source for terahertz radiation [5]. In field emission, a strong static field bends the potential at the surface of a nanoscale tip so that electrons are emitted by quantum tunneling. The apex of the tip is much smaller than the wavelength of the optical radiation, so the time-dependent electric field of the radiation is superimposed on the applied static field to modulate the current of the emitted electrons. The nonlinear current–voltage characteristics of field emission cause the emitted current to have components at frequencies that are harmonics and mixer terms relative to the incident optical radiation. The tip can be used as an antenna or transmission line to transmit electromagnetic radiation at one or more of the frequencies which are present in the current. Now, we describe simulations of laser-assisted field emission with pulsed optical radiation to determine the effects of the experimental parameters on the time for the emitted current to respond to the pulsed radiation. These simulations were made to determine the feasibility of obtaining a sub-femtosecond response—a speed three
ARTICLE IN PRESS 850
M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
orders of magnitude greater than what we have previously considered. 2. Numerical methods for simulations We have developed two different numerical methods for simulating laser-assisted field emission with pulsed optical radiation [6,7]. It is well known that simple attempts to extend the finite difference techniques, which are satisfactory for solving the time-independent Schro¨dinger equation, become numerically unstable when they are applied to solve the time-dependent Schro¨dinger equation [8]. The cause for this instability is that the numerical approximations of the operators are not unitary even though the exact operators are unitary. Split-operator formulations have been used to obtain numerical stability [9], but then the solution is only implicit so that it is necessary to form and invert a large matrix. A variety of other techniques have also been developed to achieve numerical stability [10,11] but they all have much lower computational efficiency than a direct explicit solution. Numerical implementations often require empirically adjusted ‘‘windowing’’ and ‘‘gobbling’’ to achieve numerical stability [12] and absorbing boundary conditions, in which complex potentials are added at the boundaries to limit unphysical reflected waves originating at the boundaries of the grid [13]. Both of our new procedures are numerically stable, and neither requires the use of split-operators, windowing, gobbling, or absorbing boundary conditions. In the first procedure [6], the solution for the wave function is written as the product of the solution of the time-independent Schro¨dinger equation with the pulsed radiation field deleted, and an unknown function of the spatial coordinates and time. This form for the solution is substituted into the time-dependent Schro¨dinger equation to obtain a simpler equation that is solved using finite differences to determine the unknown product function. In the second procedure [7], the solution for the wave function is written as the sum of three terms. Each of these terms contains a different unknown function of the spatial coordinates and time. The second and third terms also have products of ejot and ejot, respectively, where o is the fundamental frequency of the sinusoid that is pulsed. This solution is substituted into the time-dependent Schro¨dinger equation, which is then partitioned to give three simpler equations, which are solved using finite differences to determine the three unknown functions. The simulations in the following section of this paper were made by using the second numerical method [7] with the Fowler–Nordheim model [14] for the potential at the surface of tungsten metal. 3. Results of simulations Figs. 1(a)–(f) show the probability density for electrons near a half-space of tungsten as a function of the distance from the surface (0–3.5 nm) and the time
(0–10 fs) for laser-assisted field emission. In each of these figures ES represents the applied electrostatic field, l and P are the wavelength and power flux density of the optical radiation, and ED is the RMS value of the dynamic electric field of the radiation. While the trajectories of the electrons are not defined, one may consider the movement of the electrons by viewing the planes for consecutively larger values of time in each of these six figures. Values of the current, corresponding to the number of electrons passing a point in one second, may also be determined from these figures. Two different effects of the optical radiation may be seen in the figures. First, the dynamic electric field of the optical radiation is superimposed on the static potential barrier to cause an effective barrier height that varies to follow each cycle of the radiation. Secondly, quanta may be absorbed or emitted by the electrons to cause sudden changes in the electron energy. A sinusoidal optical pulse at a wavelength l ¼ 400 nm is turned on at the time t ¼ 0. The applied static field ES ¼ 5.5 V/nm, and the radiation field has a power flux density P ¼ 1012, 1013, and 1014 W/m2 in Figs. 1(a)–(c), respectively. These three figures show that as the power flux density is increased, there is a change in the character of the response by the electrons. For a power flux density of 1012 W/m2 or less, the magnitude of the current follows the instantaneous value of the radiation field. That is, during each cycle of the optical radiation, the current is increased when the effective height of the barrier is lowered by superimposing the electric field of the optical radiation, and decreased when the effective height of the barrier is raised. However, for a power flux density of 1014 W/m2 or greater, a massive burst of current is caused by the optical radiation, which we refer to as ‘‘catastrophic laser-assisted field emission’’. A transition between these two types of response occurs at a power flux density of 1013 W/m2, where both of these aspects are present. Figs. 1(a)–(c) also show that, even though the sine wave for the incident optical pulse starts at time t ¼ 0, there is a dead-time before this radiation causes the height of any part of the surface to be above the curve that represents the probability density for t ¼ 0. In Section 4 we show that this dead-time is associated with the duration of quantum tunneling. These three figures show that the dead-time is reduced and the rate of rise for the current is increased by increasing the power flux density. The threshold for laser ablation with thin tungsten films corresponds to a fluence of 400 J/m2 [15]. We have used a power flux density of 1016 W/m2 in the simulations for Figs. 1(d)–(f) because this is below the threshold for ablation with pulse durations of 40 fs or less. However, it is possible that the tip could still be destroyed by explosive field emission because of the additional effects of the applied static field and the resulting current. Figs. 1(d)–(f) show the probability density for electrons near a half-space of tungsten when the sinusoidal optical pulse has a power flux density P ¼ 1016 W/m2. The applied
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
851
Fig. 1. Probability density of electrons near a planar surface of tungsten (a) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1012 W/m2, ED ¼ 0.027 V/nm rms. (b) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1013 W/m2, ED ¼ 0.087 V/nm rms. (c) ES ¼ 5.5 V/nm, l ¼ 400 nm, P ¼ 1014 W/m2, ED ¼ 0.27 V/nm rms. (d) ES ¼ 5.5 V/ nm, l ¼ 400 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms. (e) ES ¼ 3.0 V/nm, l ¼ 400 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms. (f) ES ¼ 3.0 V/nm, l ¼ 1000 nm, P ¼ 1016 W/m2, ED ¼ 2.7 V/nm rms.
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
852
static field ES ¼ 5.5, 3.0, and 3.0 V/nm, and the wavelength of the optical radiation l ¼ 400, 400, and 1000 nm, respectively, in Figs. 1(d)–(f). 4. Discussion 4.1. Relevance of tunneling time Measurements of the tunnel conductance in semiconductor heterostructures [16] and experiments with Josephson junctions [17] suggest that a specific time is required for the process of quantum tunneling. The duration of quantum tunneling has considerable practical significance. For example, measurements of the tunnel conductance in heterostructures differs from theory by as much as two orders of magnitude unless the image corrections are adjusted to allow for the traversal time [18]. Many different theoretical procedures have been used to determine the time that is required to traverse a barrier by quantum tunneling [19], such as in field emission. While there is not yet a complete consensus, 12 of these methods of analysis give the same result, corresponding to the imaginary value for the time that would be calculated for this process using classical physics, which we refer to as the ‘‘semiclassical traversal time’’ [20]. Furthermore, this result is consistent with experimental measurements of the tunnel conductance in heterostructures [21]. The semiclassical traversal time is given by Z z2 dz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , T SC ¼ (1) 2 z1 m ðV ðzÞ E Þ where the particle with mass m and energy E tunnels through a potential V(z) from z1 to z2 [20]. Using the Fowler–Nordheim approximation for the potential in field emission [14] in Eq. (1), it may be shown that 0 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 2T 0 F 0 F B 2 1 F0 C qffiffiffiffiffiffiffiffiffiffiffiffiffiA, T SC ¼ (2) 1 þ 1 E@ F0 p F 1þ 1 F F0
where T 0 ¼
pffiffiffiffi 2 me 80 f3=2
(3)
4p0 f2 . (4) e3 Here e is the charge of an electron, j is the work function of the metal, e0 is the permittivity of free-space, F is the applied static field, and E(arg) is the Elliptic Integral of the Second Kind [22]. For Foo F0, which is a good approximation in field emission, these equations may be simplified to give 2T 0 F 0 T SC . (5) p F and
F0 ¼
Eq. (2) shows that for field emission with tungsten, the semiclassical traversal time is 1.4 fs for an applied static
field of 5.5 V/nm. This is in good agreement with the initial dead-time of approximately 1.5 fs that is seen in Fig. 1(a). We have already noted that the current is alternately increased and decreased during each half-cycle as the effective height of the barrier is lowered and raised by the optical radiation. However, it may be seen in Fig. 1(a) that each of these responses is delayed by approximately 0.7 fs, which is one-half of the semiclassical traversal time. 4.2. Catastrophic laser-assisted field emission By examining the data in Figs. 1(a)–(f), it may be seen that the initial dead-time is 1.5 fs for 1012 W/m2, 1.2 fs for 1013 W/m2, 0.8 fs for 1014 W/m2, and 0.4 fs in the three figures with 1016 W/m2, which shows that the dead-time is reduced as the power flux density is increased. Furthermore, these figures show that the current is greatly increased when the power flux density exceeds 1013 W/m2, which we refer to as ‘‘catastrophic laser-assisted field emission’’. Figs. 1(e) and (f) show that the power flux density has a much greater affect than the wavelength when using short pulses, and Figs. 1(d) and (e) show that reducing the applied static field causes a much sharper increase in the current. These phenomena may be understood in terms of two types of interaction: (1) the superposition of the electric field of the optical radiation on the potential barrier, and (2) the absorption and emission of quanta by the electrons. 5. Conclusions The simulations show that the dead-time before the current responds to the incident radiation may be reduced to 0.4 fs by using a greater power flux density of 1016 W/m2, where the current is increased to the point that we call ‘‘catastrophic laser-assisted field emission’’. This is several orders of magnitude faster than the phenomena that we have considered before [5], and these results suggest that it may be possible to use this method for an ultrafast optoelectronic switch. We have also found that the dead-time, as well as other delays in the response to the radiation, may be explained in terms of the semiclassical traversal time for quantum tunneling within the potential barrier. Experiments that are pertinent to these simulations are now in progress. Others have generated electron pulses with durations of tens of fs by irradiating a metal field emission tip with a fs laser [23,24], and they have reported a transition from photofield emission to optical field emission that is similar to the one that we have shown in these simulations at a much shorter time scale [23]. Analyses of recent measurements in which sub-8 fs lasers irradiated tungsten field emission tips show that this technique has promise for producing a single electron pulse with a duration of less than 1 fs [25]. Our simulations have the delimitations of (1) simulating the response of a half-space of metal so that three-dimensional
ARTICLE IN PRESS M.J. Hagmann, M.S. Mousa / Ultramicroscopy 107 (2007) 849–853
effects are not included, and (2) simulating the incident radiation as a sinusoidal field that is switched on whereas others have used the model of a wave packet with a Gaussian envelope which more accurately models the propagation of a pulse.
[8] [9] [10] [11] [12]
Acknowledgment
[13] [14]
The authors are grateful to Dr. E. P. Sheshin and Dr. A. S. Baturin, both of the Moscow Institute of Physics and Technology, Moscow, Russia, and M. Brugat, of Image Instrumentation Inc., Cooper City, FL, USA, for their advice concerning the technology of field emission. References [1] [2] [3] [4] [5] [6] [7]
P.B. Corkum, et al., Opt. Lett. 19 (1994) 1870. T.W. Ha¨nsch, Opt. Commun. 80 (1990) 71. J.J. Macklin, et al., Phys. Rev. Lett. 70 (1993) 766. R. Lopez-Martens, et al., Phys. Rev. Lett. 94 (2005) 33001. M.J. Hagmann, IEEE Trans. Microw. Theory Tech. 52 (2004) 236. M.J. Hagmann, Int. J. Quant. Chem. 60 (1996) 1231. M.J. Hagmann, Int. J. Quant. Chem. 70 (1998) 703.
[15] [16] [17] [18] [19] [20] [21]
[22]
[23] [24] [25]
853
A. Goldberg, et al., Am. J. Phys. 35 (1967) 177. K.C. Kulander, Phys. Rev. A 35 (1987) 445. C.W. McCurdy, et al., Phys. Rev. A 43 (1991) 5980. M.R. Hermann, J.A. Fleck Jr., Phys. Rev. A 38 (1988) 6000. J.T. Muckerman, et al., in: C. Cerjan (Ed.), Numerical Grid Methods and their Applications to Schro¨dinger’s Equation, Academic, Dordrecht, 1992. A. Vibok, G.G. Balint-Kurti, J. Chem. Phys. 96 (1992) 7615. R. Gomer, Field Emission and Field Ionization, American Institute of Physics, New York, 1993, pp. 53–54. Z. Toth, et al., Appl. Phys. A: Mat. Sci. Process. 60 (1995) 431. P. Gueret, et al., Appl. Phys. Lett. 53 (1988) 1617. D. Esteve, et al., Phys. Scr. T 29 (1989) 121. P. Gueret, et al., Electron. Lett. 21 (1985) 344. R. Landauer, Th. Martin, Rev. Mod. Phys. 66 (1994) 217. M.J. Hagmann, Int. J. Quant. Chem. 44 (1992) 299. P. Gueret, in: J.M. Chamberlain, et al., (Eds.), Electronic Properties of Multilayers and Low-Dimensional Semiconductor Structures, Plenum, New York, 1990, pp. 317–329. M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government Printing Office, Washington, 1972, pp. 589–592. P. Hommelhoff, et al., Phys. Rev. Lett. 96 (2006) 077401. B.N. Mironov, et al., JETP Lett. 83 (2006) 369. P. Hommelhoff, et al., arXiv:quant-ph/0607006 v1 2 Jul 2006.